The invention is based on a fuel injection pump as generically defined hereinafter. In a known fuel injection pump of this kind U.S. Pat. No. 8,712,763, the driver members are embodied as clamping rings, which are adjustably fixed on the round governor rod by clamping the free arms together using a screw, and which have a tang that engages the control slide in an annular groove provided for that purpose. As a result, although the individual driver members can be adjusted relative to one another with respect to their rotational position and hence the control slides can be adjusted relative to one another with respect to their stroke position, they cannot be actuated independently of one another. Thus it is likewise impossible for a partial shutoff of the engine cylinders during idling, which is particularly desirable in large engines, to be attained as in another known fuel injection pump (German Pat. No. 28 21 161), by shutting off the pumping of fuel from one portion of the pump work chambers to the engine. These provisions primarily make it possible to improve the fuel consumption and exhaust and noise emissions during engine idling and partial-load operation. Also, less fuel gets into the motor oil to cause excessive thinning of the oil. Further, a cylinder shutoff is also used in motor vehicles whenever only accessory equipment needs to be driven by the vehicle engine, such as when loading or unloading silo or tank trucks. In contrast to these known in-line injection pumps, in which the pump pistons of individual pump units are rotatable in order to shut pumping down to zero pumping, this provision has not been known in slide-controlled pumps up to now, because a rotation of the slide causes a change in the fuel quantity only if a corresponding oblique-edge control is provided between the relief bore and the control slide.